Finite Group Databasebeta

← #6436687

Subgroups — #6436687

Order 512.

1813 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3#35326C211yesmaximal, characteristic
4C2×#2320C211yesmaximal, characteristic, complemented
5#29875C212yesmaximal, complemented
6#29840C212yesmaximal, complemented
7#29837C212yesmaximal, complemented
8#29805C212yesmaximal, complemented
9#29761C212yesmaximal
10#29757C212yesmaximal, complemented
11#29719C212yesmaximal
12#29715C212yesmaximal, complemented
13#23695C214yesmaximal, complemented
14#23691C214yesmaximal, complemented
15#21394C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic
17#20814C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3117—28no—
22#3104—28no—
23C2×#484C2211yescharacteristic
24C2×#484C2212yes—
25C2×#481C2211yescharacteristic
26C2×#479C2212yes—
27C2×#477C2212yes—
28C2×#464C2212yes—
29C2×#462C2212yes—
30C2×#451C2211yescharacteristic
31C2×#451C2212yes—
32C2×#448C2211yescharacteristic
33C2×#441C2212yes—
34C2×#440C2212yes—
35C2×C4×Q16C2214yes—
36C2×#432C2211yescharacteristic
37C2×#431C2211yescharacteristic
38#2857C2212yes—
39#2850C2212yes—
40#2660C2211yescharacteristic
41#2658C2211yescharacteristic
42#2642C2211yescharacteristic
43#2640C2211yescharacteristic
44#2611—21no—
45#2611—21no—
46#2604—22no—
47#2564—22no—
48#2498—21no—
49#2498—21no—
50#2327C2212yes—
51#2326C2212yes—
52#2320—22no—
53#2320—21no—
54#2320—21no—
55#2320C2212yes—
56#2320C2214yes—
57#2320C2212yes—
58C2×#391C2212yes—
59C2×#389C2212yes—
60C2×#387—22no—
61C2×#383—21no—
62C2×#382—21no—
63C2×C4×C4⋊C4C2211yescharacteristic
64C2×#376C2211yescharacteristic
65#2085C2214yes—
66#2083C2214yes—
67#2074C2214yes—
68#2072C2214yes—
69#2068C2214yes—
70#2068C2214yes—
71#2067C2214yes—
72#2066C2214yes—
73#2056C2214yes—
74#2056C2214yes—
75#2056C2214yes—
76#2055C2214yes—
77#2043C2214yes—
78#2042C2214yes—
79#2016C2214yes—
80#2015C2214yes—
81#1989C2212yes—
82#1988C2212yes—
83#1983C2212yes—
84#1981C2212yes—
85#1874C2212yes—
86#1872C2212yes—
87#1872C2212yes—
88#1872C2212yes—
89#1843C2212yes—
90#1832C2212yes—
91#1797C2212yes—
92#1797C2212yes—
93#1768C2214yes—
94#1763C2214yes—
95C4×#103C2212yes—
96C4×#102C2212yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2314yes—
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101#533—24no—
102#533—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106#529—24no—
107#529—24no—
108C2×#126—21no—
109C2×#125—21no—
110C2×#125C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#124C2312yes—
113C2×#123—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2311yescharacteristic
117C2×#121C2311yescharacteristic
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—24no—
122C2×C4×Q8D418yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#484—28no—
140#484—24no—
141#481—24no—
142#479—28no—
143#477—28no—
144#464—28no—
145#462—28no—
146#451—28no—
147#451—24no—
148#448—24no—
149#441—28no—
150#440—28no—
151C4×Q16—216no—
152#432—24no—
153#431—24no—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#106C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161C2×#103C2314yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×C22⋊C8C2311yescharacteristic
172C2×C22⋊C8C2311yescharacteristic
173C2×C22⋊C8C2312yes—
174C2×#97C2312yes—
175C2×C4×C8C2312yes—
176#400—22no—
177#400—21no—
178#400—21no—
179#399—22no—
180#399—21no—
181#399—21no—
182#398—21no—
183#398—21no—
184#397—21no—
185#397—21no—
186#395—22no—
187#393—22no—
188#391—22no—
189#391—22no—
190#391C2314yes—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#391C2314yes—
195#390C2312yes—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389C2314yes—
201#389C2314yes—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—24no—
209#387—24no—
210#387—24no—
211#387—24no—
212#387—22no—
213#387—22no—
214#386—22no—
215#386—22no—
216#385—22no—
217#385—22no—
218#384C2312yes—
219#383—24no—
220#383—22no—
221#383—22no—
222#382—24no—
223#382—22no—
224#382—22no—
225#382C2312yes—
226#380—22no—
227#380—22no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232#376D412yes—
233#376D411yescharacteristic
234#376D411yescharacteristic
235#376C2312yes—
236#376C2311yescharacteristic
237#376C2311yescharacteristic
238C2×#95—21no—
239C2×#95—22no—
240C2×#95—22no—
241C2×#95—22no—
242C2×#95D412yes—
243C2×#95D411yescharacteristic
244C2×#95D411yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—44no—
255C22×Q8C2×D414yes—
256C22×Q8C2411yescharacteristic
257C22×Q8C2412yes—
258C22×Q8C2412yes—
259C22×D4—21no—
260C22×D4C2411yescharacteristic
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4—21no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2×D412yes—
268C23×C4C2×D411yescharacteristic
269C23×C4C2411yescharacteristic
270C2×Q16—28no—
271C2×Q16—24no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×SD16—24no—
276C2×SD16—44no—
277C22×C8C2412yes—
278C22×C8C2414yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—44no—
285#126—44no—
286#125—22no—
287#125—22no—
288#125—44no—
289#124—22no—
290#124—22no—
291#124—44no—
292#123—44no—
293C22⋊Q8—22no—
294C22⋊Q8—24no—
295C22⋊Q8—22no—
296C22⋊Q8—24no—
297C22⋊Q8—44no—
298#121—22no—
299#121—22no—
300C4×Q8—28no—
301C4×Q8—28no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—22no—
305C4×Q8—24no—
306C4×Q8—22no—
307C4×Q8—24no—
308C4×Q8—44no—
309C4×Q8C2×D418yes—
310C4×Q8C2×D418yes—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—44no—
314#117—44no—
315#117—44no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C42—28no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42—82no—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D414yes—
369C2×C42#4114yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—28no—
375#107—24no—
376#106—24no—
377C4⋊C8—24no—
378C4⋊C8—24no—
379C4⋊C8—44no—
380C4⋊C8—44no—
381#103—28no—
382#103—24no—
383#103—24no—
384#103—28no—
385#103—24no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—44no—
391C22⋊C8—24no—
392C22⋊C8—22no—
393C22⋊C8—22no—
394#97—24no—
395C4×C8—24no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95C2×D412yes—
413#95C2×D411yescharacteristic
414#95C2×D411yescharacteristic
415#95C2×D412yes—
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D412yes—
419#95C2×D412yes—
420#95C2×D412yes—
421#95C2×D412yes—
422C24—21no—
423C24C22×D411yescharacteristic
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—48no—
436C2×Q8—44no—
437C2×Q8—48no—
438C2×Q8—42no—
439C2×Q8—81no—
440C2×Q8—81no—
441C2×Q8C22×D414yes—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—24no—
445C2×D4—42no—
446C2×D4—44no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4#12114yes—
477C22×C4#12114yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4112yes—
489C22×C4C2511yescharacteristic
490Q16—48no—
491Q16—416no—
492SD16—48no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C22×D414yes—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—28no—
540C42—28no—
541C42—24no—
542C42—24no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—44no—
547C42—82no—
548C42—82no—
549C42#12114yes—
550C42#12114yes—
551C42C22×D412yes—
552C42C22×D412yes—
553C42C2×#4112yes—
554C42C2×#4112yes—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23C2×#12112yes—
564C23#57814yes—
565C23C23×D411yescharacteristic
566Q8—24no—
567Q8—24no—
568Q8—48no—
569Q8—44no—
570Q8—42no—
571Q8—44no—
572Q8—42no—
573Q8—44no—
574Q8—84no—
575D4—42no—
576D4—42no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4#58514yes—
636C8—42no—
637C8—44no—
638C8—42no—
639C22—22no—
640C22—24no—
641C22—24no—
642C22—22no—
643C22—81no—
644C22—81no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22#304414yes—
649C22C22×#12111yescharacteristic
650C22#344912yes—
651C22C2×#57812yes—
652C22#358511yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—44no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—44no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—84no—
669C4—81no—
670C4—82no—
671C4#344912yes—
672C4#345512yes—
673C2—24no—
674C2—161no—
675C2#3463311yescharacteristic
676C2C2×#304411yescharacteristic
677C2#6042511yescharacteristic
678C2#6098112yes—
679C2#6098312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#3104C2semidirect
4. C2×#2320C2semidirect
5. #29875C2semidirect
6. #29840C2semidirect
7. #29837C2semidirect
8. #29805C2semidirect
10. #29757C2semidirect
12. #29715C2semidirect
13. #23695C2semidirect
14. #23691C2semidirect
15. #21394C2semidirect
17. #20814C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#31172
C2×#31042
#353262
C2×#23202
#298752
#298402
#298372
#298052
#297612
#297572
#297192
#297152
#236952
#236912
#213942
#213882
#208142